What about while the engine is running? We need this data. Disconnect any accessories wired to the main battery and check voltage accross the terminals while engine is running.
12v is low for a healthy, charged battery at rest.
Shorted cells in a battery should cause voltage drop, rather than increased voltage, but who knows!?
If the alternator is overcharging, it is possible to have a residual surface charge for a short time. You can turn on the lights for a minute or two to dissipate surface voltage after shutting off the engine; but it doesn't seem like that is your issue.
Do you have any capacitors in your stereo or lighting systems?
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